1. Let A A = {: C: Im ***21} - 2 (a) Sketch the set A in the complex plane. (b) is := -2i a boundary point of A? Provide

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1 Let A A C Im 21 2 A Sketch The Set A In The Complex Plane B Is 2i A Boundary Point Of A Provide 1 (41.94 KiB) Viewed 31 times

1. Let A A = {: C: Im ***21} - 2 (a) Sketch the set A in the complex plane. (b) is := -2i a boundary point of A? Provide reasons for your answer. (c) Is this set open, closed, both or neither? Provide reasons for your answer. 2. Suppose f2) = 4, Write in the form (3) = u(,y) + (x,y), where = = x + iy and u and v are real-valued functions. 3. Let (2) 87 (a) Calculate limf(a) using properties of limits. (b) Use the - definition of continuity to show that is continuous at : = 1. 4. Calculate limusing properties of limits. 5. Use the definition of a derivativo (see p.55,56) to show that I'() = 2: for all : € C, if (=) = +1. 6. Let 9() = 2.5 (a) Use the Cauchy-Riemann equations to determine where g is differentiable. (b) Is g analytic anywhere? Justify your answer. 7. Let S(=) = . Use the polar form of the Cauchy-Riemann equations to determine where / is differentiable. 8. Let g(x,y) = y (a) Show that, is harmonic in D = {(x,y): +(+3) < 4). (1) How can the function (3) = be used to show that y is harmonic in D? 9. Show that ce etw for any , w EC.